Simplify and expand the following expression: $ \dfrac{2}{4y - 36}- \dfrac{5}{y - 5}+ \dfrac{1}{y^2 - 14y + 45} $
First find a common denominator by finding the least common multiple of the denominators. Try factoring the denominators. We can factor a $4$ out of denominator in the first term: $ \dfrac{2}{4y - 36} = \dfrac{2}{4(y - 9)}$ We can factor the quadratic in the third term: $ \dfrac{1}{y^2 - 14y + 45} = \dfrac{1}{(y - 9)(y - 5)}$ Now we have: $ \dfrac{2}{4(y - 9)}- \dfrac{5}{y - 5}+ \dfrac{1}{(y - 9)(y - 5)} $ The least common multiple of the denominators is: $ 4(y - 9)(y - 5)$ In order to get the first term over $4(y - 9)(y - 5)$ , multiply by $\dfrac{y - 5}{y - 5}$ $ \dfrac{2}{4(y - 9)} \times \dfrac{y - 5}{y - 5} = \dfrac{2(y - 5)}{4(y - 9)(y - 5)} $ In order to get the second term over $4(y - 9)(y - 5)$ , multiply by $\dfrac{4(y - 9)}{4(y - 9)}$ $ \dfrac{5}{y - 5} \times \dfrac{4(y - 9)}{4(y - 9)} = \dfrac{20(y - 9)}{4(y - 9)(y - 5)} $ In order to get the third term over $4(y - 9)(y - 5)$ , multiply by $\dfrac{4}{4}$ $ \dfrac{1}{(y - 9)(y - 5)} \times \dfrac{4}{4} = \dfrac{4}{4(y - 9)(y - 5)} $ Now we have: $ \dfrac{2(y - 5)}{4(y - 9)(y - 5)} - \dfrac{20(y - 9)}{4(y - 9)(y - 5)} + \dfrac{4}{4(y - 9)(y - 5)} $ $ = \dfrac{ 2(y - 5) - 20(y - 9) + 4} {4(y - 9)(y - 5)} $ Expand: $ = \dfrac{2y - 10 - 20y + 180 + 4}{4y^2 - 56y + 180} $ $ = \dfrac{-18y + 174}{4y^2 - 56y + 180}$ Simplify: $ = \dfrac{-9y + 87}{2y^2 - 28y + 90}$